3.67.80 \(\int e^{-7+x^2-25^{\frac {4 x}{e^2}} (x^2)^{\frac {4 x}{e^2}}} (e^2 (1+2 x^2)+25^{\frac {4 x}{e^2}} (x^2)^{\frac {4 x}{e^2}} (-8 x-4 x \log (25 x^2))) \, dx\)

Optimal. Leaf size=29 \[ e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \]

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Rubi [F]  time = 1.68, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*(E^2*(1 + 2*x^2) + 25^((4*x)/E^2)*(x^2)^((4*x)/E^2)*(-8*x
- 4*x*Log[25*x^2])),x]

[Out]

Defer[Int][E^(-5 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2)), x] + 2*Defer[Int][E^(-5 + x^2 - 25^((4*x)/E^2)*(x^
2)^((4*x)/E^2))*x^2, x] - 8*Defer[Int][25^((4*x)/E^2)*E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*x*(x^2)^
((4*x)/E^2), x] - 4*Log[25*x^2]*Defer[Int][25^((4*x)/E^2)*E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*x*(x
^2)^((4*x)/E^2), x] + 8*Defer[Int][Defer[Int][25^((4*x)/E^2)*E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*x
*(x^2)^((4*x)/E^2), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (1+2 x^2\right )-4\ 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \left (2+\log \left (25 x^2\right )\right )\right ) \, dx\\ &=-\left (4 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \left (2+\log \left (25 x^2\right )\right ) \, dx\right )+\int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (1+2 x^2\right ) \, dx\\ &=-\left (4 \int \left (2\ 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}}+25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \log \left (25 x^2\right )\right ) \, dx\right )+\int \left (e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}}+2 e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x^2\right ) \, dx\\ &=2 \int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x^2 \, dx-4 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \log \left (25 x^2\right ) \, dx-8 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx+\int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \, dx\\ &=2 \int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x^2 \, dx+4 \int \frac {2 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx}{x} \, dx-8 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx-\left (4 \log \left (25 x^2\right )\right ) \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx+\int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \, dx\\ &=2 \int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x^2 \, dx-8 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx+8 \int \frac {\int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx}{x} \, dx-\left (4 \log \left (25 x^2\right )\right ) \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx+\int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.43, size = 29, normalized size = 1.00 \begin {gather*} e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*(E^2*(1 + 2*x^2) + 25^((4*x)/E^2)*(x^2)^((4*x)/E^2)*
(-8*x - 4*x*Log[25*x^2])),x]

[Out]

E^(-5 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*x

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fricas [A]  time = 0.60, size = 21, normalized size = 0.72 \begin {gather*} x e^{\left (x^{2} - \left (25 \, x^{2}\right )^{4 \, x e^{\left (-2\right )}} - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*log(25*x^2)-8*x)*exp(2*x*log(25*x^2)/exp(2))^2+(2*x^2+1)*exp(2))*exp(-exp(2*x*log(25*x^2)/exp
(2))^2+x^2-5)/exp(2),x, algorithm="fricas")

[Out]

x*e^(x^2 - (25*x^2)^(4*x*e^(-2)) - 5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -{\left (4 \, {\left (x \log \left (25 \, x^{2}\right ) + 2 \, x\right )} \left (25 \, x^{2}\right )^{4 \, x e^{\left (-2\right )}} - {\left (2 \, x^{2} + 1\right )} e^{2}\right )} e^{\left (x^{2} - \left (25 \, x^{2}\right )^{4 \, x e^{\left (-2\right )}} - 7\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*log(25*x^2)-8*x)*exp(2*x*log(25*x^2)/exp(2))^2+(2*x^2+1)*exp(2))*exp(-exp(2*x*log(25*x^2)/exp
(2))^2+x^2-5)/exp(2),x, algorithm="giac")

[Out]

integrate(-(4*(x*log(25*x^2) + 2*x)*(25*x^2)^(4*x*e^(-2)) - (2*x^2 + 1)*e^2)*e^(x^2 - (25*x^2)^(4*x*e^(-2)) -
7), x)

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maple [A]  time = 0.62, size = 24, normalized size = 0.83

method	result	size
risch	\(x \,{\mathrm e}^{-\left (25 x^{2}\right )^{4 x \,{\mathrm e}^{-2}}+x^{2}-5}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x*ln(25*x^2)-8*x)*exp(2*x*ln(25*x^2)/exp(2))^2+(2*x^2+1)*exp(2))*exp(-exp(2*x*ln(25*x^2)/exp(2))^2+x^
2-5)/exp(2),x,method=_RETURNVERBOSE)

[Out]

x*exp(-((25*x^2)^(2*x*exp(-2)))^2+x^2-5)

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maxima [A]  time = 0.58, size = 26, normalized size = 0.90 \begin {gather*} x e^{\left (x^{2} - e^{\left (8 \, x e^{\left (-2\right )} \log \relax (5) + 8 \, x e^{\left (-2\right )} \log \relax (x)\right )} - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*log(25*x^2)-8*x)*exp(2*x*log(25*x^2)/exp(2))^2+(2*x^2+1)*exp(2))*exp(-exp(2*x*log(25*x^2)/exp
(2))^2+x^2-5)/exp(2),x, algorithm="maxima")

[Out]

x*e^(x^2 - e^(8*x*e^(-2)*log(5) + 8*x*e^(-2)*log(x)) - 5)

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mupad [B]  time = 4.52, size = 21, normalized size = 0.72 \begin {gather*} x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{-{\left (390625\,x^8\right )}^{x\,{\mathrm {e}}^{-2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x^2 - exp(4*x*exp(-2)*log(25*x^2)) - 5)*exp(-2)*(exp(4*x*exp(-2)*log(25*x^2))*(8*x + 4*x*log(25*x^2))
- exp(2)*(2*x^2 + 1)),x)

[Out]

x*exp(x^2)*exp(-5)*exp(-(390625*x^8)^(x*exp(-2)))

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sympy [A]  time = 36.83, size = 22, normalized size = 0.76 \begin {gather*} x e^{x^{2} - e^{\frac {4 x \log {\left (25 x^{2} \right )}}{e^{2}}} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*ln(25*x**2)-8*x)*exp(2*x*ln(25*x**2)/exp(2))**2+(2*x**2+1)*exp(2))*exp(-exp(2*x*ln(25*x**2)/e
xp(2))**2+x**2-5)/exp(2),x)

[Out]

x*exp(x**2 - exp(4*x*exp(-2)*log(25*x**2)) - 5)

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